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2.1 Obtain the following derivatives from first principles: (a) the first derivative of 3x + 4; (b) the first, second and third derivatives of x- + x; (c) the first derivative of sin x. 2.2 Find from first principles the first derivative of (x + 3) and compare your answer with that obtained using the chain rule. 2.3 Find the first derivatives of (a) x expx. (b) 2 sin x cosx, (c) sin 2x. (d) x sinax. (e) (expax)(sin ax) tan ' ax, (f) In(x" + x-"), (g) In(a* +a * ), (h) x. 2.4 Find the first derivatives of (a) x/(a + x), (b) x/(1 - x)'/2, (c) tanx, as sin x/ cosx, (d) (3x3 + 2x + 1)/(8x2 - 4x + 2). 2.5 Use result (2.12) to find the first derivatives of (a) (2x + 3) ', (b) sec' x, (c) cosech'3x, (d) 1/ Inx, (e) 1/[sin '(x/a)]. 2.6 Show that the function y(x) = exp(-(x)) defined by exp x for x 0, is not differentiable at x = 0. Consider the limiting process for both Ax > 0 and Ax 2.13 Show that the lowest value taken by the function 3x + 4x3 - 12x2 + 6 is -26. 2.14 By finding their stationary points and examining their general forms, determine the range of values that each of the following functions y(x) can take. In each case make a sketch-graph incorporating the features you have identified. (a) y(x) = (x -1)/(x2 + 2x +6). (b) (x) =1/(4 +3x -x-). (c) v(x) = (8 sin x)/(15 + 8 tan x). 2.15 Show that y(x) = xa"exp x' has no stationary points other than x = 0. if exp(-/2)